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arxiv: 2510.26252 · v2 · submitted 2025-10-30 · 🧮 math.RT · math.AC· math.AG· math.RA

Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

Ryu Tomonaga This is my paper

Pith reviewed 2026-05-18 03:52 UTC · model grok-4.3

classification 🧮 math.RT math.ACmath.AGmath.RA
keywords non-commutative crepant resolutionstoric singularitiesdivisor class groupIyama-Wemyss mutationslattice polytopesdimer modelsGorenstein toric singularitiesupper sets
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The pith

Toric NCCRs of Gorenstein toric singularities with rank-one divisor class group are classified by non-trivial upper sets in a quotient of the class group under a partial order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies toric non-commutative crepant resolutions for Gorenstein toric singularities where the divisor class group has rank one. These resolutions correspond one-to-one with non-trivial upper sets in a quotient of the divisor class group that carries a specific partial order. This correspondence proves that all such resolutions are connected by sequences of Iyama-Wemyss mutations and are therefore derived equivalent. The classification also yields explicit quivers with relations via higher-dimensional dimer models and confirms that the number of indecomposable summands equals the normalized volume of an associated lattice polytope.

Core claim

We prove the existence and give a classification of toric non-commutative crepant resolutions of Gorenstein toric singularities with divisor class group of rank one. They correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order, and all such NCCRs are connected by iterated Iyama-Wemyss mutations.

What carries the argument

The partial order on a quotient of the divisor class group, whose non-trivial upper sets biject with the toric NCCRs.

If this is right

  • All toric NCCRs for these singularities are derived equivalent.
  • The quivers with relations of the NCCRs can be described using higher-dimensional dimer models.
  • The number of indecomposable direct summands of a toric NCCR equals the normalized volume of the corresponding lattice polytope.
  • There is an explicit formula for the volume of d-dimensional lattice polytopes with d+2 vertices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This combinatorial approach via upper sets might generalize to cases with higher rank class groups if suitable orders can be defined.
  • The mutation connectivity suggests that these resolutions form a single derived equivalence class, potentially simplifying computations in related categories.
  • The volume verification supports broader conjectures on the structure of NCCRs in toric geometry.

Load-bearing premise

That the chosen partial order on the quotient of the divisor class group and its upper sets precisely correspond to all possible toric non-commutative crepant resolutions without omissions or extras.

What would settle it

Finding a Gorenstein toric singularity with divisor class group of rank one possessing a toric NCCR that does not correspond to any non-trivial upper set in the defined quotient, or two NCCRs not linked by Iyama-Wemyss mutations.

read the original abstract

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. This classification allows us to prove that all toric NCCRs of such toric singularities are connected by iterated Iyama-Wemyss mutations, and hence are derived equivalent to each other. We also describe explicitly the quivers with relations of these toric NCCRs in terms of higher dimensional analogue of dimer models. In the Appendix, we give an explicit formula for the volume of $d$-dimensional lattice polytopes with $d+2$ vertices. As an application, we verify a conjecture of Van den Bergh in the case of Gorenstein toric singularities with divisor class group of rank one: the number of indecomposable direct summands of a toric NCCR coincides with the normalized volume of the corresponding lattice polytope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the existence and gives a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. These NCCRs correspond bijectively to non-trivial upper sets in a quotient of the divisor class group equipped with a partial order induced from the toric fan data and the Gorenstein condition. The classification is used to show that all such NCCRs are connected by iterated Iyama-Wemyss mutations and hence derived equivalent. The associated algebras are constructed explicitly as higher-dimensional dimer models. The appendix supplies an independent volume formula for d-dimensional lattice polytopes with d+2 vertices and applies it to verify Van den Bergh's conjecture that the number of indecomposable summands equals the normalized volume of the corresponding polytope.

Significance. If the combinatorial correspondence holds, the paper delivers a complete, parameter-free classification together with mutation connectivity and an explicit dimer-model realization for this restricted class of toric singularities. The independent volume formula in the appendix and the resulting verification of the conjecture constitute additional strengths that make the results falsifiable and directly applicable. These contributions advance the understanding of NCCRs in toric geometry and noncommutative algebraic geometry.

minor comments (3)
  1. [Main construction (around the poset definition)] The definition of the partial order on the quotient of the divisor class group is central; a brief remark on its independence from the choice of supporting hyperplane (beyond the Gorenstein condition) would improve readability in the main construction section.
  2. [Appendix] In the appendix, the volume formula is derived combinatorially; including a short explicit computation for a 3-dimensional polytope with 5 vertices would help readers verify the formula before the conjecture application.
  3. [Section on quiver descriptions] Notation for the higher-dimensional dimer models could be clarified by adding a small diagram or table comparing the 2-dimensional and higher-dimensional cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the results, and recommendation to accept the manuscript. The referee's description correctly captures the classification of toric NCCRs via upper sets in the quotient of the divisor class group, the mutation connectivity, the dimer-model realization, and the appendix verification of Van den Bergh's conjecture.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the partial order on the quotient of the divisor class group directly from the toric fan data and the Gorenstein condition via an induced height function, constructs the NCCR algebras explicitly as higher-dimensional dimer models, and proves the bijective correspondence to non-trivial upper sets by direct verification that every such upper set yields an NCCR and conversely. Mutation connectivity follows from the poset structure without external reduction. The appendix volume formula is derived independently and applied to confirm a count conjecture. No load-bearing step reduces by definition, fitting, or self-citation chain to its own inputs; the derivation is self-contained combinatorial geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from toric geometry and representation theory of algebras; no new free parameters are introduced and no new entities are postulated beyond the combinatorial objects already present in the literature on NCCRs.

axioms (2)
  • standard math Standard properties of Gorenstein toric singularities and their divisor class groups
    The setup assumes the usual definitions and exact sequences relating the class group to the lattice and the fan.
  • standard math Existence of Iyama-Wemyss mutation functors for NCCRs
    The mutation connectedness statement invokes the general theory of mutations developed by Iyama and Wemyss.

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    3, 813–857

    Claire Amiot, Osamu Iyama, and Idun Reiten,Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday, American Journal of Mathematics137 (2015), no. 3, 813–857

    work page 2015

  2. [2]

    1, 277–301

    Lev Borisov and Zheng Hua,On the conjecture of King for smooth toric Deligne–Mumford stacks, Advances in Math- ematics221(2009), no. 1, 277–301

    work page 2009

  3. [3]

    Nathan Broomhead,Dimer models and Calabi-Yau algebras, American Mathematical Society215(2012), no. 1011. 16 RYU TOMONAGA

    work page 2012

  4. [4]

    2, 572–618

    Aslak Bakke Buan, Bethany Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov,Tilting theory and cluster combinatorics, Advances in mathematics204(2006), no. 2, 572–618

    work page 2006

  5. [5]

    Norihiro Hanihara,Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary represen- tation type, arXiv:2303.14625, 2023

    work page arXiv 2023

  6. [6]

    ,Higher hereditary algebras arising from some toric singularities, arXiv:2412.19040, 2024

    work page arXiv 2024

  7. [7]

    Norihiro Hanihara and Osamu Iyama,Enhanced Auslander–Reiten duality and tilting theory for singularity categories, arXiv:2209.14090, 2022

    work page arXiv 2022

  8. [8]

    Wahei Hara,Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops, Advances in mathematics318(2017), 355–410

    work page 2017

  9. [9]

    Akihiro Higashitani and Nakajima Yusuke,Conic divisorial ideals of hibi rings and their applications to non- commutative crepant resolutions, Selecta Mathematica25(2019), no. 5, 78

    work page 2019

  10. [10]

    1, 51–82

    Osamu Iyama,Auslander correspondence, Advances in mathematics210(2007), no. 1, 51–82

    work page 2007

  11. [11]

    1, 22–50

    ,Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Advances in mathematics 210(2007), no. 1, 22–50

    work page 2007

  12. [12]

    4, 1087–1149

    Osamu Iyama and Idun Reiten,Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras, American Journal of Mathematics130(2008), no. 4, 1087–1149

    work page 2008

  13. [13]

    3, 521–586

    Osamu Iyama and Michael Wemyss,Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Inventiones mathematicae197(2014), no. 3, 521–586

    work page 2014

  14. [14]

    181, American Mathematical Soc., 2012

    Graham Joseph Leuschke and Roger Wiegand,Cohen-Macaulay representations, Graduate Texts in Mathematics, no. 181, American Mathematical Soc., 2012

    work page 2012

  15. [15]

    Koji Matsushita,Conic divisorial ideals of toric rings and applications to Hibi rings and stable set rings, (2022)

    work page 2022

  16. [16]

    41, Birk˘ auser, 1983

    Richard Peter Stanley,Combinatorics and commutative algebra, Progress in Math, vol. 41, Birk˘ auser, 1983

    work page 1983

  17. [17]

    Ryu Tomonaga,Higher hereditary algebras and toric Fano stacks with Picard number at most two, in preparation

    work page

  18. [18]

    ,Weak del Pezzo surfaces yield2-hereditary algebras and3-Calabi-Yau algebras, in preparation

    work page

  19. [19]

    ,Cohen-Macaulay representations of invariant subrings, arXiv:2403.19282, 2024

    work page arXiv 2024

  20. [20]

    Berlin, Heidelberg: Springer Berlin Heidelberg (2004), 749–770

    Michel Van den Bergh,Non-commutative crepant resolutions, The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002. Berlin, Heidelberg: Springer Berlin Heidelberg (2004), 749–770

    work page 2002

  21. [21]

    J.122(2004), no

    ,Three-dimensional flops and noncommutative rings, Duke Math. J.122(2004), no. 3, 423–455

    work page 2004

  22. [22]

    ˇSpela ˇSpenko and Michel Van den Bergh,Non-commutative resolutions of quotient singularities for reductive groups, Inventiones mathematicae210(2017), 3–67

    work page 2017

  23. [23]

    ,Non-commutative crepant resolutions for some toric singularities i, International Mathematics Research No- tices21(2020), 8120–8138

    work page 2020

  24. [24]

    1, 73–103

    ,Non-commutative crepant resolutions for some toric singularities ii, Journal of Noncommutative Geometry14 (2020), no. 1, 73–103

    work page 2020

  25. [25]

    2, 435–521

    Michael Wemyss,Flops and clusters in the homological minimal model programme, Inventiones mathematicae211 (2018), no. 2, 435–521

    work page 2018

  26. [26]

    146, Cambridge University Press, 1990

    Yuji Yoshino,Cohen-Macaulay modules over Cohen-Macaulay rings, vol. 146, Cambridge University Press, 1990. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan Email address:ryu-tomonaga@g.ecc.u-tokyo.ac.jp

    work page 1990